Nonlinear Equations - UFRJ

x
FOREWORD
The following is unknown:
Conjecture F.2 (P ≠ NP). There cannot possibly exist an algorithm
that decides problem F.1 in at most O(S r ) operations, for any
fixed r > 1.
Above, an algorithm means a Turing machine, or a discrete RAM
machine. For references, see [42]. Problem F.1 is AN9 p.251. It is
still NP-hard if the degree of each monomial is ≤ 2.
In these notes we are mainly concerned about equations over the
field of complex numbers. There is an analogous problem to 4-SAT
(see [42]) or to Problem F.1, namely:
Problem F.3 (HN2, Hilbert Nullstellensatz for degree 2). Given a
system of complex polynomials f = (f 1 , . . . , f s ) ∈ C[x 1 , . . . , x n ], each
equation of degree 2, decide if there is x ∈ C n with f(x) = 0.
The polynomial above is said to have size S = ∑ S i where S i is the
number of monomials of f i . The following is also open (I personally
believe it can be easier than the classical P ≠ NP).
Conjecture F.4 (P ≠ NP over C). There cannot possibly exist an
algorithm that decides HN2 in at most O(S r ) operations, for any fixed
r > 1.
Here, an algorithm means a machine over C and I refer to [20]
for the precise definition.
We are not launching an attack to those hard problems here
(see [63] for a credible attempt). Instead, we will be happy to obtain
solution counts that are correct almost everywhere, or to look for
algorithms that are efficient on average.